Ocean Engineering 38 (2011) 2185–2200

Contents lists available at SciVerse ScienceDirect

Ocean Engineering

0029-80

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/oceaneng

Computation of wave loads on the superstructures ofcoastal highway bridges

Jun Jin a,n, Bo Meng b

a Dept. of Maritime Systems Engineering, Texas A&M University at Galveston, P.O. Box 1675, Galveston, TX 77553, USAb Naval Architect, FloaTEC, LLC, 14701 St. Mary’s lane, Suite 250, Houston, TX 77079, USA

a r t i c l e i n f o

Article history:

Received 20 August 2010

Accepted 27 September 2011

Editor-in-Chief: A.I. Incecikpushed off the substructure by storm surge and wave action. To prevent such structural failure of

coastal bridges, wave loads on bridge superstructures need to be quantified. In this study, two different

Available online 20 October 2011

Keywords:

Bridge design

Bridge loads

Wave loads

Wave action

Computational fluid dynamics

18/$ - see front matter & 2011 Elsevier Ltd. A

016/j.oceaneng.2011.09.029

esponding author.

ail addresses: jinj@tamug.edu (J. Jin), bmeng@

a b s t r a c t

Recent hurricanes have caused severe structural damages to a number of coastal highway bridges along

the US coast of the Gulf of Mexico. A large amount of bridge superstructure sections were displaced or

numerical models were used to analyze wave-structure interaction and compute wave loads.

Computational Fluid Dynamics (CFD) Software Flow-3D was used to analyze the effects of green water

loading and superstructure elevation on wave forces. A 2D potential flow model was developed for

computation of wave loads on bridge superstructures fully submerged in water. Wave induced pressure

on bridge superstructures was obtained by solving complex velocity potential from Laplace’s equation

and boundary conditions using finite difference method. The 2D potential flow model was validated by

large-scale laboratory measurements, and then used to perform parametric study using a range of wave

parameters, water depth and bridge superstructure width. Equations for calculating wave loads on

bridge superstructures were developed using results of the parametric study.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Background

In recent years, several hurricanes have caused structuraldamages to coastal highway bridges in all four states along theUS coast of the Gulf of Mexico. In September 2004, duringHurricane Ivan, the 2.5 mile long I-10 twin bridges over EscambiaBay near Pensacola Florida suffered extensive structural damages.Fifty-eight spans of the eastbound and westbound bridges werepushed off the substructure and another sixty-six spans weremisaligned (Talbot, 2005). A new bridge with much higher spanswas completed in 2007 in a $243 million project (Buckley, 2006).In August 2005, during Hurricane Katrina, 44 highway bridgeswere damaged in Alabama, Mississippi and Louisiana (ASCETCLEE, 2006). Three bridges were completely damaged in asimilar way as the I-10 bridges over Escambia Bay, includingthe I-10 twin span across Lake Pontchartrain in Louisiana, theBiloxi Bay Bridge and Bay St. Louis Bridge in Mississippi. The twobridges in Mississippi were both parts of US 90. Repair costs wereestimated as $30 million, $275 million and $276 million for thesethree bridges (Padgett et al., 2008). In September 2008, during

ll rights reserved.

floatec.com (B. Meng).

Hurricane Ike, two bridges in Galveston, Texas, the Rollover PassBridge and the Pelican Island Bridge were partially damaged.

The four coastal highway bridges that were completely damagedduring recent hurricanes had the same structural form as shown inFig. 1. The beam and slab structural form is the most commonstructural form of coastal highway bridges due to its simple andfaster construction (Rajagopalan, 2006). The superstructure of abeam and slab bridge consists of parallel pre-stressed concretegirders connected longitudinally by concrete slab and may also beconnected transversely by diaphragms. The substructure consists ofconcrete piles connected on top by pile caps. The superstructure isplaced on top of steel bearing plates, which are placed under eachgirder and are bolted to the pile caps through anchor bolts.

During a hurricane, the storm surge combines with the normaltide to create the ‘‘hurricane storm tide’’, which may raise themean water level to have the superstructure exposed to waveactions. Examination of damaged bridges suggested that the fail-ure of these bridges was caused by weak connection between thesuperstructure and the substructure, which was merely providedby anchoring bolts. Shear failure of anchoring bolts was observedin damaged bridges (OEA, 2005; Eamon et al., 2007; Okeil and Cai,2008). Wave loads easily broke the weak connection and pushedthe superstructure progressively during the passage of each wave.

To prevent wave induced structural damage to coastal bridges,either in design of new bridges or retrofitting of existing bridges,the magnitude of wave loads on bridge superstructures must be

Fig. 1. Structure of beam-and-slab type highway bridge. (a) Definition of struc-

tural elements and (b) Structure of Pelican Island Bridge in Galveston, Texas.

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–22002186

determined with reasonable accuracy. In general, computation ofwave loads on structures is complicated by the complex nature ofwave�structure interaction. Because structure in water inter-rupts the propagation of waves, the flow field associated withwaves depends on both wave conditions and geometry of thestructure.

1.2. Existing methods for calculation of wave loads on bridge

superstructure

Douglass et al. (2006) reviewed methods for estimating waveloads on highway bridge superstructures and other types ofcoastal and offshore structures, and stated that ‘‘existing litera-ture is not adequate for estimating wave loads on bridge decks forthe design of highways.’’ They suggested that the method ofMcConnell et al. (2004) ‘‘is the most appropriate to adapt to thehighway bridge deck problem,’’ despite noting that the methodwas based on physical model tests on a small scale jetty head,which was quite different from a highway bridge superstructure.The method of McConnell et al. (2004) was later modified byCuomo et al. (2007). In a preliminary application to the U.S. 90bridge across Biloxi Bay, Chen et al. (2009) used this method toestimate the wave loads on bridge superstructures. Because themethod was developed for beam and deck elements of a jettyhead, it was necessary for Chen et al. (2009) to divide the bridgesuperstructure into beam and deck components, estimate themaximum wave load on each component and then combine thesemaxima. It may be expected that this approach will lead toconsiderable error because the maximum wave load on thecomponents does not occur at the same time.

AASHTO recently published Guide Specifications for Bridges

Vulnerable to Coastal Storms (AASHTO, 2008), which include

equations for computation of wave loads on bridge superstruc-tures. According to Marin and Sheppard (2009), these equationswere developed on the basis of Morison’s equation (Morison et al.,1950) and Kaplan’s equations (Kaplan, 1992; Kaplan et al., 1995).

Morison’s equation is widely applied in design of offshore andcoastal structures for structures with length in wave direction (e.g.diameter for cylindrical members) relatively small compared withthe wave length, such as piles, braces and legs of jacket platforms.The semi-empirical equation was based on the assumption thatwave kinematics is not significantly affected by the presence ofstructures. Wave loads can then be calculated as the sum ofacceleration dependent inertia force and velocity dependent dragforce. Both inertia and drag forces are modified by empiricalcoefficients determined by laboratory model tests. For offshorestructures, application of Morison’s equation is limited to struc-tures with diameter smaller than 1/5 of wave length (API, 2000). InCoastal Engineering Manual (USACE, 2002), the limit on diameter ofstructure is reduced to 1/20 of wavelength.

Kaplan’s equations were developed for determination of waveimpact forces on flat plate-like horizontal deck structures thatwere above still water level (SWL) but lower than wave crest. Thedeck was assumed to be stationary and the sea water could goover and around the plate. The impact force was considered as acombination of momentum and drag forces. Waves were assumedto be large compared to the dimensions of platform deck; there-fore, wave kinematics was considered not affected by the presenceof platform deck. Spatial variations of wave kinematics along thedeck were also neglected. The deck was also assumed to be verythin; therefore, buoyancy force was neglected in the equations.

Due to the large size of bridge superstructure and complicatedgeometry, the interaction between structure and waves shouldnot be neglected. Therefore it may not be appropriate to useKaplan’s equations to determine wave loads on bridge super-structures. Instead, diffraction analysis needs to be performed tocompute wave induced pressure on the surface of superstruc-tures. Wave loads can then be computed by integrating pressureon wetted area of superstructures. There are many analytical andnumerical studies on the diffraction of waves by submerged orfloating structures. These structures are used as breakwaters incoastal area and have simple cross-section, either cylindrical orrectangular. Structures studied are commonly assumed to beinfinitely long; therefore the problem can be solved by 2D modelsin the x�z plane. A comprehensive review of these studies can befound in Zheng et al. (2007).

1.3. Existing experimental research on wave loads on bridge

superstructures

After Hurricanes Ivan and Katrina, a few experimental studieswere conducted to investigate wave loads on bridge superstruc-tures. Three studies reviewed here were conducted in the US atTexas A&M University at College Station (McPherson, 2008),Oregon State University (Bradner, 2008; Schumacher et al.,2008) and University of Florida (Marin and Sheppard, 2009).One study was conducted in Japan (Cuomo et al., 2009).

In the TAMU study (McPherson, 2008), the physical model wasa 1:20 scale model of the superstructure of Biloxi Bay Bridge madefrom acrylic. The model was suspended at a fixed height. Forcesacting on the model were measured by a single force transducer.The physical model in the OSU study (Bradner, 2008) was a 1:5scale model of the previous I-10 bridge over Escambia Bay, Florida.The model was made from reinforced concrete. Wave loads on themodel were measured by six load cells, two for horizontal loadsand four for vertical loads. Pressure on the surface of structure wasmeasured by 13 pressure transducers. Two different physicalmodel setups were tested, rigid and flexible. In the rigid setup,

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–2200 2187

the model was fixed to the support. In the flexible setup, themodel was connected to springs in the horizontal direction. Thephysical model used in the UF study (Marin and Sheppard, 2009)was a 1:8 scale model of the previous I-10 bridge over EscambiaBay. Wave loads on the model were measured by four 3-compo-nent load cells. Experimental results of the UF study were found inAASHTO (2008). Cuomo et al (2009) conducted testing on a 1:10scale model consisted of four girders. Pressure under the slab andbottom of girders was measured by pressure transducers. All fourstudies conducted tests with a range of wave parameters (heightand period) and water depth.

Observations in these experimental studies are summarizedhere.

1.

The role of initial wave impact:Bea et al. (1999) separated wave force on a platform deck intobuoyancy force, slamming force, drag force, lift force andinertia force. The impulsive slamming force reflected thetransfer of momentum from water to structures. It wasfollowed by the sum of other force components, which devel-oped as the wave inundates the deck. The peak of slammingforce was much higher than the peak of the slowly varyingwave inundation force. This force history pattern was observedin experimental studies that measured pressure under thestructure (Cuomo et al., 2009).In the OSU study, Bradner (2008) compared simultaneouspressure and force measurements. Pressure data showed theimpulse and slow-varying pattern in some cases. However, allof the load cell data showed only the slowly varying force.Measured wave force in the TAMU study (McPherson, 2008)and the UF study (AASHTO, 2008) also showed no impulsivewave force. Due to the spatial variation of pressure on thesurface of bridge superstructures, it is not reasonable toestimate wave force on the entire structure from pressuremeasurements at only a few locations. Therefore, the study ofCuomo et al (2009) will not be further discussed in this paper.

2.

Horizontal wave load vs. vertical wave load:The length of bridge superstructure in wave direction is muchhigher than the height (see Fig. 1). As a result, the verticalcomponent of wave load should be higher than the horizontalcomponent. In both the OSU study (Bradner, 2008) and theTAMU study (McPherson, 2008), peak vertical loads werefound to be 4–6 times as large as the corresponding peakhorizontal load. Only three pairs of vertical and horizontalforce measurements of the UF study were found (AASHTO,2008). The peak vertical loads were respectively 13, 18 and 20times as large as corresponding peak horizontal loads. Thesevalues were quite different from results of the other twostudies. We noticed that measured horizontal loads were verysmall (less than 10 lbs).

3.

Wave loads vs. water surface elevation:In the TAMU study (McPherson, 2008), water surface elevationright in front of the physical model was measured. Timehistory of vertical wave load and surface elevation in front ofthe model indicated that vertical wave load was completely in-phase with surface elevation. In the OSU study (Schumacheret al., 2008), for rigid setup, it was observed that maximumhorizontal wave load occurred at the same time as maximumvertical wave load.

4.

Wave loads vs. wave parameters:McPherson (2008) tested regular waves with combinations ofthree wave heights and three wave periods and observed thatvertical wave load increased with increasing wave height andwave period. Bradner (2008) tested regular waves with com-binations of six wave heights and six wave periods and at fivedifferent water levels. It was observed that both horizontal and

vertical wave loads consistently increased with increasingwave height. The effect of wave period on wave loads wasnot consistent at different water levels. A clear trend wasobserved only when the model was fully submerged. Thehorizontal load decreased with increasing wave period, whilethe vertical load increased with increasing wave period.

When wave crest is higher than the top surface of bridgesuperstructure, water is able to overtop the superstructure, whichgenerates downward acting force on the structure. The loadsgenerated by water on deck are generally known as ‘‘green waterloading’’ in Naval Architecture. None of the experimental studiesinvestigated the effect of green water loading. In addition,because the models were all tested at fixed elevation in thesestudies, the effect of deck elevation on wave loads was notinvestigated.

1.4. Objective and scope

At present the complex wave�structure interaction may bewell simulated by full 3D Navier�Stokes solvers. For example,COBRAS (Lin and Liu, 1998) has been used for modeling of waveinteraction with submerged permeable structures (Lara et al.,2006). Similarly, the commercial CFD code Flow-3D (Flow Science,2008) has been used for simulating the intricate flow field detailsaround submarine pipeline (Smith and Foster, 2005) and bridgepiers (Richardson and Panchang, 1998). Thus, this approach maybe suitable for investigating wave�structure interaction andcomputing wave loads on bridge superstructures. However, it isunreasonable to expect a bridge engineer to make these simula-tions because these models are extremely complex and requirespecialized skill.

Our objective is to develop easy to use tools for computation ofwave loads on the superstructures of coastal bridges, which willassist bridge designers to design new coastal bridges and/orretrofit existing bridges to resist wave loads. To achieve this, wewill make a very large number of numerical simulations thatcover a range of wave parameters and bridge geometry.

Section 2 describes analysis of wave kinematics on the freesurface using nonlinear wave theory, and evaluation of Kaplan’sequations (Kaplan, 1992; Kaplan et al., 1995). Section 3 describesanalyses we performed using CFD software Flow-3D (FlowScience, 2008). In Section 4, formulation of the 2D potential flowmodel was presented. Section 5 describes validation of the 2Dpotential flow model using Flow-3D and large-scale test results.Section 6 presents the results of the parametric study weperformed using the 2D potential flow model. Equations wedeveloped for calculation of maximum wave loads on bridgesuperstructures are given in Section 7. Summary and conclusionsare presented in Section 8.

2. Theoretical analysis of wave loads on bridgesuperstructures

For a better understanding of the interaction between hurri-cane waves and superstructures of coastal bridges, we used wavetheory to calculate kinematics of a hurricane wave. We used thewave conditions at US 90 bridge over Biloxi Bay, Mississippi,during Hurricane Katrina in the analysis. Storm Surge and waveconditions near the bridge during Hurricane Katrina were hind-casted by Chen et al. (2009). At the peak of storm surge, thewater depth was 8.75 m, the maximum wave height was 4.68 m,peak wave period was 5.5 s and the bottom of the highestbridge superstructure that collapsed was 0.19 m above still waterlevel (SWL).

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–22002188

According to Le Mehaute (1976), suitable wave theory for thewave was the Stokes wave theory at 4th order approximation. Weused the Fenton 5th order solution (Fenton, 1985) to computewater surface elevation, water particle velocity and accelerationin x (horizontal) and z (vertical) directions. The wave wasassumed to be traveling in the x direction, normal to the longspan of bridge superstructure. The wave profiles of linear waveand Stokes 5th order wave are plotted in the x�z plane in Fig. 2,together with water particle velocity and acceleration on the freesurface at the same time. The cross section of the superstructureis plotted in Fig. 2 with its center placed below wave crest. Crestelevation of the wave was 3.14 m, 34% higher than the waveamplitude. Length of the wave was 41.1 m, portion of wave aboveSWL was 15.62 m long, larger than the width of the super-structure (10.16 m). However, because the bottom of the super-structure was 0.19 m above SWL, the Stokes 5th order wave couldnot have the superstructure fully immersed.

Fig. 3. Temporal variations of w

Fig. 2. Wave profiles and spatial variations of wave kinematics parameters. (a) Wa

parameters of Stokes 5th order wave.

To investigate phase differences between wave kinematicsparameters, time history of these parameters at a fixed x valuewas divided by their maxima. As shown in Fig. 3, only horizontalvelocity was in phase with surface elevation. Vertical velocity andhorizontal acceleration were approximately in phase and wereahead of surface elevation by about p/3. They reached theirmaxima when water surface was up-crossing SWL and rapidlydropped to zero at wave crest. Vertical acceleration and surfaceelevation were in antiphase. Maximum vertical accelerationoccurred at the trough and was negative above SWL.

Kaplan (1992) gave the following equation for total vertical waveload on a flat plate:

Fz ¼ rp8

c2 €Zþrp4_Zc@c

@tþr2_Z _Z�� ��cCD

� �b ð1Þ

where r is the density of water, c is the wetted length, b is the widthof the plate, Z is the surface elevation and CD is the drag coefficient.

ave kinematics parameters.

ve profiles of linear wave and Stokes 5th order wave and (b) Wave kinematics

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–2200 2189

The first two terms in Eq. (1) define the momentum compo-nent; the third term is drag force. The first term is proportional tovertical acceleration, which remains negative while wave is risingabove SWL. The second term is proportional to vertical velocity.This term increases from the time when waves reach the leadingedge of the plate to the time when waves reach the trailing edgeof the plate. When the plate is fully submerged, the second termdrops to zero. The drag force is a result of flow separation.Therefore, it remains zero before the plate is fully submerged.

We used Eq. (1) to calculate time history of vertical wave loads ona piece of deck plate between two girders. The width of the plate was1.43 m, and was set 1 m above SWL. As shown in Fig. 4a, maximumupward wave load was determined by the first two terms in Eq. (1),which occurred when water reached the trailing end of plate. Afterthat, the vertical load was composed of drag and inertia forces andkept decreasing even when water was rising. We also applied Eq. (1)to calculate vertical wave load on a 10.2-m wide plate (Fig. 4b).Maximum vertical load was 367 kN, compared to 305 kN for the1.43 m wide plate, not proportional to plate width.

We concluded that it is not appropriate to use Kaplan’sequation (Kaplan, 1992; Kaplan et al., 1995) to compute verticalwave load on bridge superstructure. The reasons are given below.

1.

Kaplan’s equation used kinematics of undisturbed waves tocalculate wave load. The underlying assumption was that the

Fig. 4. Time history of vertical forces calculated by Kaplan’s equation

dimensions of structures were relatively small compared towavelength. Because of the large size of bridge superstruc-tures, wave kinematics was significantly affected. In addition,the parallel girders also cause flow complexities as wavesimpact the superstructure, which should not be neglected.

2.

Kaplan’s equation gave downward acting force while wavewas impacting the structure from below. As shown in Fig. 4,total vertical load turned negative before the wave reached itscrest. This was caused by the vertical acceleration, which wasnegative above SWL (see Fig. 3).

3.

Kaplan’’s equation gave decreasing vertical load while watersurface was rising. This conflicts with the observation fromexperimental studies. Time history of vertical wave load andsurface elevation in front of the model indicated that verticalwave load was completely in-phase with surface elevation(McPherson, 2008).

3. Computation of wave loads on bridge superstructure usingFlow-3D

We used commercial Computational Fluid Dynamics (CFD) soft-ware Flow-3D (Flow Science, 2008) to analyze wave and structureinteraction and compute wave loads on bridge superstructures.

(Kaplan, 1992). (a) Width¼1.43m and (b) Width¼10.2m.

Fig. 6. Variation of volume of fluid during Flow-3D wave simulation.

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–22002190

Flow-3D is capable of solving the full 3D Navier�Stokes equationsusing the volume of fluid (VOF) method. The VOF method enablesmodeling of the free surface and liquid/gas interface. We firstvalidated the capability of Flow-3D to model Stokes nonlinearwaves. Then we investigated the effects of superstructure elevationand green water loading on wave loads.

3.1. Validation of Flow-3D for modeling of Stokes nonlinear waves

We used two approaches to validate the capability of Flow-3D tomodel Stokes nonlinear waves, volume of fluid and water particlekinematics. The computational domain (Fig. 5) was 10 m high and200 m long. Water depth was set at 6 m. The domain was dis-cretized into 50,000 0.2�0.2 m2 grids in the x�z plane. The waterwas seawater at 20 1C, and the density was 1.027 kg/m3.

As shown in Fig. 5, boundary AB was defined as incoming waveboundary. Nonlinear waves with 2 m wave height and 6 s periodwere defined as inflow boundary condition on AB. Boundary CDwas defined as outflow boundary. During the simulation, volumeof the water varied with the propagation of waves. Variation ofvolume was computed and plotted in Fig. 6 for 10 wave periods.Wave celerity was 6.8 m/s, it took 29.4 s for the first wave toreach the outflow boundary. As shown in the figure, volume offluid increased as waves entered the computation domain. At29.4 s, when the first wave reached the outflow boundary, volumeof fluid returned to its original level and oscillated around thislevel afterwards. Maximum volume variation was less than 1.5%.

To examine water particle kinematics, we selected three points(E, F and G in Fig. 5) in the computation domain as particlesources. These points were located at 5 m from the bottom (1 mbelow SWL). Distances from these points to incoming boundarywere 10, 95 and 180 m respectively. During the simulation, eachparticle sources released 10 particles per second. The particletrajectories were plotted in Fig. 7. Particle trajectories beneath aStokes wave are not closed orbits. The higher order terms inparticle displacement equations drag the particles toward thedirection of wave propagation. As a result, particle trajectories arelooping orbits drifting in the direction of wave propagation. Asshown in Fig. 7, all particles drifted forward with the waves.Water particle velocities of point E were plotted in Fig. 8. Bothhorizontal and vertical velocities agreed well with Stokes 5thorder theory. We concluded that water particle kinematics agreedwith the Stokes nonlinear wave theory.

3.2. Simulation of wave structure interaction using Flow-3D

After Flow-3D was validated, we added bridge superstructureto the simulation. We used the geometry of the superstructure ofold I-10 twin bridges over Escambia Bay, Pensacola Florida(Fig. 9). The bridges were damaged during Hurricane Ivan in2004. The I-shaped girders were simplified as rectangular sec-tions, bridge rails were neglected for simplification. We usedStokes 5th order theory for input wave.

Fig. 5. Computational domain and pa

3.2.1. Wave loads vs. superstructure elevation

To investigate the relationship between wave loads and super-structure elevation, we set bridge superstructure at eleven dif-ferent elevation levels. Input wave height was 2 m, wave periodwas 6 s and water depth was 6 m. Green water loading wascomputed in addition to the resultant vertical wave load on theentire superstructure. Maximum values of wave loads wereplotted vs. superstructure elevation in Fig. 10. Time history ofwave loads was plotted in Fig. 11 for four cases, with top surfaceat 0.8 m above SWL, at SWL, at 0.8 m below SWL and 2 m belowSWL. Wave profile and pressure contours at selected time wereplotted in Fig. 12.

As shown in Fig. 10, maximum horizontal wave load was lesssensitive to superstructure elevation than maximum verticalwave load. When the top surface of the superstructure was higherthan wave crest, the maximum horizontal wave load started todecrease with superstructure elevation. Maximum vertical waveload was approximately invariant to superstructure elevationwhen the superstructure was lower than the trough (0.76 m)and decreased with superstructure elevation when the super-structure was above the trough.

While the superstructure was near SWL, maximum verticalloads were about 4–6 times as large as corresponding maximumhorizontal loads. Vertical loads and horizontal loads were also inphase in these cases (see Fig. 11a and b). The relationshipbetween vertical and horizontal wave loads agrees with observa-tions of experimental studies (Bradner, 2008; McPherson, 2008).

3.2.2. Green water loading vs. vertical wave loads

As shown by Fig. 10, green water loading decreased signifi-cantly with superstructure elevation. To investigate the effect ofwave heights on green water loading, we performed four addi-tional simulations using waves with 6 s period and wave heights

rticle source points in Flow-3D.

Fig. 7. Particle trajectories and pressure variation during Flow-3D wave simulation (Pressure unit: pa; length unit: m).

Fig. 8. Comparison of computed particle velocities of point E with theoretical

results (Velocity unit: m/s, time unit: s).

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–2200 2191

of 1, 3, 4 and 4.6 m. Top surface of the superstructure was set atSWL for these simulations. The 4.6 m wave was a near-breakingwave. Maximum wave loads were plotted vs. wave height inFig. 13. Wave profile and pressure contours at the time ofmaximum vertical load are plotted in Fig. 14.

The maximum value of green water loading was relativelyhigh compared to corresponding maximum vertical wave load.The maximum value of green water loading exceeded 50% ofmaximum vertical wave load for waves 2 m or higher. However,time history of wave loads (Fig. 11b) indicated that maxima ofgreen water loading and vertical wave load occurred at differenttimes. Wave profile at the time of maximum vertical load(Figs. 12c and 14) indicated that maximum vertical wave forceoccurred when wave crest was right in front of the superstruc-ture, there was little water on the top surface at that time. Weremoved green water loading from vertical wave load to evaluatethe difference made by green water on vertical wave load. Themaximum values of vertical load without green water loading are

also plotted in Fig. 13. As shown in Fig. 13, the effect of greenwater loading increased with wave height. The difference madeby green water loading in vertical wave load was 10% for 1 mwave, and was 23% for 4.6 m wave.

3.2.3. Flow field around bridge superstructure

Flow field around bridge superstructure was illustrated byvectors of water particle velocity in Fig. 15. As shown in thefigure, flow field was interrupted by the girders. Within the openchambers formed by girders and superstructure, water flow wasalong the boundaries in a circular pattern. It was also observedthat flow complexity increased with increasing wave height.

The following were conclusions we made from Flow-3Dsimulation results:

1.

Vertical wave force on bridge superstructure decreases withincreasing superstructure elevation, while horizontal waveforce is not significantly affected when top surface of bridgesuperstructure is below wave crest.

2.

Green water loading on bridge superstructure increases withwave height. For near breaking wave, maximum green waterloading exceeded 50% of maximum vertical wave force. How-ever, due to the phase differences between green waterloading and upward wave forces, the maximum value ofvertical wave force was not significantly reduced from upwardwave force by green water loading.

3.

Flow field around bridge superstructures was interrupted byparallel girders; therefore, methods based on wave kinematicsshould not be used to compute wave forces.

Our objective is to develop easy to use tools for computation ofwave forces on the superstructure of coastal bridges, which willcover a range of wave parameters and bridge geometry. Toachieve this, we will make a very large number of simulations.Although Flow-3D was able to model nonlinear waves and wavebreaking/overtopping, the computation time of each simulationwas very long. Adding the time and effort required in setting upthe model and processing simulation results made it difficult toperform the large number of simulations that are needed todevelop a design tool.

As an alternative, we considered the potential flow theory asdescribed below. Potential flow theory was based on assumptionsof inviscid and incompressible fluids and irrotational flow. Themotion of fluids was defined by the velocity potential, which wasgoverned by Laplace’s equation. Potential flow theory has beenused for analyzing interaction between waves and offshore andcoastal structures. For example, WAMIT (Wamit, 2008) has beenused for analyzing wave interaction with floating and submergedoffshore structures. Potential flow theory was not able to modelwave breaking and green water loading. However, Flow-3Dresults showed that maximum vertical wave force was notsignificantly affected by green water loading.

Fig. 9. Dimensions of the superstructures of old I-10 bridges over Escambia Bay.

Fig. 10. Maximum wave forces vs. deck elevation.

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–22002192

4. Formulation of the 2D potential flow model

The problem was considered as diffraction of a regular linearmonochromatic wave by bridge superstructure in water of uni-form depth d. A 2D Cartesian coordinate system was defined asshown in Fig. 16. The structure was considered rigid and sta-tionary. Assuming inviscid and incompressible fluid and irrota-tional flow, the motion of water was described by the velocitypotential:

jðx,z,tÞ ¼ ReðFe�iotÞ ð2Þ

where F(x,z) is the time-independent complex velocity potential,which satisfies the Laplace’s equation in the fluid domain(Fig. 17):

r2F¼@2F@x2þ@2F@z2¼ 0 ð3Þ

The complex velocity potential F is composed of two parts, theincident wave potential FI and the diffracted wave potential FD,and satisfies the following boundary conditions.

On the free surface:

@F@z�o2

gF¼ 0 ð4aÞ

At sea-bed:

@F@z¼ 0 ð4bÞ

On the surface of structure:

@F@n¼ 0 ð4cÞ

Incoming boundary at x¼�N:

@F@x�ik 2FI�F½ � ¼ 0 ð4dÞ

Outgoing boundary at x¼N

@F@x�ikF¼ 0 ð4eÞ

The incident wave with wave height H and period T is travelingin the positive x direction and has the following complex incidentpotential:

FI ¼�igH

2ocoshkðzþdÞ

coshkdeikx ð5Þ

where o¼2p/T is the circular frequency of wave, g is thegravitational acceleration and k is the wave number, which isrelated to o by the dispersion relationship:

o2 ¼ gktanhkd ð6Þ

After the boundary value problem for diffracted wave potentialFD is solved, the water surface elevation Z and hydrodynamicpressure p are determined as

Z¼�1

g

@j@t¼�

og

ImðFe�iotÞ ð7Þ

p¼�rgz�r @j@t¼�rgz�roImðFe�iotÞ ð8Þ

where r is the mass density of sea water.

4.1. Numerical solution

The problem was solved in the x�z plane by finite differencemethod. The governing equation and boundary conditions werediscretized at each grid points (Fig. 17) as follows:

Fi,j ¼14ðFi,jþ1þFi,j�1þFiþ1,jþFi�1,jÞ ð9Þ

At seabed and horizontal surface of structure:

Fi,mþ1 ¼Fi,m ð10aÞ

On the vertical surface of structure:

Fnþ1,j ¼Fn,j ð10bÞ

On the free surface:

Fi,1 2�o2

gDz

� �þFi,2 �2�

o2

gDz

� �¼ 0 ð10cÞ

Incoming boundary:

F1,jð�2þ ikDxÞþF2,jð2þ ikDxÞ ¼ 4ikDxcoshkðdþzÞ

coshkdð10dÞ

Outgoing boundary:

FI,j ¼1

1�ikDxFI�1,j ð10eÞ

The discretized equations were solved by the conjugate gra-dients method proposed by Panchang et al. (1991) to obtain thevelocity potential and the water pressure in the domain. The wave

-600-400-200

0200400600800

1,0001,2001,400

0Time (s)

Forc

e (k

N)

-1,500-1,000

-5000

5001,0001,5002,0002,5003,000

Forc

e (k

N)

5 10 15 20 25 30 35 40 0Time (s)

5 10 15 20 25 30 35 40

0Time (s)

5 10 15 20 25 30 35 400Time (s)

5 10 15 20 25 30 35 40

3,500

-1,000-500

0500

1,0001,5002,0002,5003,000

Forc

e (k

N)

3,500

-1,000-500

0500

1,0001,5002,0002,5003,000

Forc

e (k

N)

Horizontal

Green WaterVertical

HorizontalVerticalGreen Water

Fig. 11. Time history of wave forces on bridge superstructures. (a) Top surface at 0.8m above SWL, (b) Top surface at SWL, (c) Top surface at 0.8m below SWL and (d) Top

surface at 2.0m below SWL.

Fig. 12. Wave profile and pressure contours at different moments (Pressure unit: Pa). (a) Top surface at 2.0m above SWL, max. horizontal force, (b) Top surface at 2.0m

above SWL, max. vertical force, (c) Top surface at SWL, max. horizontal and vertical forces, (d) Top surface at SWL, max. green water, (e) Top surface at 0.8m below SWL,

max. vertical force and (f) Top surface at 2.0m below SWL, max. vertical force.

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–2200 2193

0

1,000

2,000

3,000

4,000

5,000

6,000

0Wave Height (m)

Forc

e (k

N)

1 2 3 4 5

Max. VerticalMax. Green WaterMax HorizontalMax Vertical w/o Green Water

Fig. 13. Maximum wave forces vs. wave height (top surface at SWL).

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–22002194

forces on the structure were then calculated by integrating thepressure on the surface of superstructure along x and z directions.Details of the numerical solution can be found in Meng (2008).

5. Validation of the 2D potential flow model

We first compared results of the 2D potential flow model toFlow-3D results. We modeled the same structure as in Flow-3Danalyses. The computation domain was defined as 6�29.4 m2,distance from incoming and outgoing boundaries to the nearestface of structure was both 10 m. Grid size was 0.1�0.1 m2, andthe domain was meshed by 17,995 grids.

Four analyses were performed. Input wave period was 6 s, andwave height was 1, 2, 3, 4 and 4.6 m. Computed maximum waveloads are plotted in Fig. 18. As shown in the figure, results of the2D potential flow model were generally higher than Flow-3Dresults. The only exception was for the near breaking wave heightof 4.6 m, the maximum vertical wave force computed by Flow-3Dwas higher. The 2D potential flow model was based on linearwave theory, while in Flow-3D analyses, the nonlinear Stokeswave theory was used. Wave crest of nonlinear wave is higherthan the crest of linear wave of the same wave height. As a result,the 2D potential flow model should give smaller wave forces thanFlow-3D. On the other hand, the 2D potential flow modelneglected wave breaking and green water loading. Therefore theoverall differences in results were not significant.

5.1. Comparison with OSU experimental results

We used the test results of OSU study (Schumacher et al., 2008)to validate the 2D potential flow model. Results presented inSchumacher et al. (2008) were slightly different from those inBradner (2008). Because bridge superstructure was treated as fixedboundary in the analytical model, we selected test results of rigidsetup for comparison. Water depth was 1.89 m, and wave periodwas 2.5 s. Five different wave heights were tested, 0.34, 0.43, 0.53,0.66 and 0.84 m, corresponding crest heights according to Stokes5th order wave theory were 0.19, 0.24, 0.31, 0.40 and 0.54 mrespectively. In the tests, the bottom of the model structure wasfixed at SWL, whereas in 2D potential flow model, the top of thestructure was at SWL. Model results were adjusted according tosuperstructure elevation. In Section 3, we presented Flow-3Dresults of maximum wave forces on superstructures with differentdeck elevation (Fig. 10). Maximum horizontal wave load was notsensitive to superstructure elevation when the top surface of thesuperstructure was below wave crest. When part of the super-structure was out of water, assuming that the horizontal wave force

was proportional to wetted depth of the superstructure(c4Zmax),Eq. (11) was derived to modify the horizontal wave force to deckelevation. Maximum vertical wave load decreased with super-structure elevation when the superstructure was above the trough.Assuming that the pressure linearly decreases from SWL to wavecrest, Eq. (12) was derived for modification of vertical wave force todeck elevation:

FhðcÞ

Fh0¼min 1,1�

c�Zmax

db

� �ð11Þ

FvðcÞ

Fv0¼ 1�

c

Zmaxþdbð12Þ

where c is the superstructure clearance, which is measured fromtop of the superstructure to SWL. Fh(c) and Fv(c) are, respectively,horizontal and vertical wave force when superstructure clearanceequals c. Fh0, Fv0 are, respectively, horizontal and vertical wave forcewhen c is 0. Zmax is crest height, db is the overall depth of bridgesuperstructure. Both horizontal and vertical forces were thenmultiplied by the ratio of 2Zmax/H to account for wave nonlinearity.

As shown in Fig. 19, both horizontal and vertical forcescomputed by the 2D potential flow model agreed well with testresults. At wave height equal to 0.34 m, differences betweenmodel results and measured results were relatively large, 18%and 16%, respectively, for horizontal and vertical forces. For otherwave heights, the differences were within 10%.

5.2. Comparison with experimental results of Cuomo et al. (2007)

Cuomo et al. (2007) conducted experimental studies on waveloadings on exposed jetties. The model was a 1:25 scale model ofa jetty head made of wood. Force and pressure under the deck andbeams were measured in the tests.

The 2D model was set up to compute wave forces on the jettyhead using wave conditions tested by Cuomo et al. (2007).Dimensions of the model are shown in Fig. 20. The top surfaceof the structure was on SWL in the 2D model. The results weremodified to three different deck clearance levels (0.06, 0.11 and0.16 m) using Eq. (12). Comparison of the quasi-static uplift forceson the external deck element was shown in Fig. 21. Measuredresults were plotted against slightly higher wave heights to avoidoverlap of data points. As shown in Fig. 21, for all wave heights,maximum values of model results and measured results agreedvery well. Differences in maximum values were, respectively,4.4%, 4.9%, 3.4% and 2.0% for Hs equal to 0.1, 0.14, 0.18 and 0.22 m.For 0.1 and 0.18 m waves, range of model results agreed well withmeasured results. For 0.14 and 0.22 m waves, range of modelresults was much larger than the range of measured results.Because measured quasi-static uplift forces in Cuomo et al. (2007)were plotted against significant wave heights, we were unable todetermine wave period and deck clearance of each force mea-surement. Therefore, we did not evaluate the differences betweenmodel results and measured results individually.

6. Parametric study using the 2D potential flow model

We used the 2D potential flow model to perform parametricstudy that covers a range of wave height, wave period, waterdepth and bridge geometry. Geometry of the superstructure of oldI-10 twin bridges over Escambia Bay, Pensacola Florida (Fig. 9)was used in the study. Range of wave period was from 2 to 10 s,with 0.2 s increment. Range of wave height was determined bythe breaking wave height. For flat bottom, the breaking waveheight is determined by the lower value of Eqs. (13) and (14)

Fig. 14. Wave profile and pressure contours at moment of maximum vertical force (Pressure unit: Pa). (a) H¼1.0m, (b) H¼3.0m, (c) H¼4.0m and (d) H¼4.6m.

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–2200 2195

(Weggel, 1972):

Hb

Lb¼ 0:142tanh

2pd

Lb

� �ð13Þ

Hb

db¼ 0:78 ð14Þ

where Hb is the breaking wave height, Lb is the wavelength ofbreaking wave and db is the water depth.

6.1. Wave loads vs. wave height and wave period

Because linear wave was used as input in the analytical model,wave loads computed were linearly related to wave heights.

Fig. 15. Vectors of water particle velocity at the moment of maximum wave force (Velocity Unit: m/s). (a) H¼1.0m, (b) H¼2.0m, (c) H¼3.0m, (d) H¼4.0m and (e) H¼4.6m.

Incident wave (H, T)

Free Surface

Inco

min

g B

ound

ary

Out

goin

g B

ound

ary

Seabed

Bridge Superstructure

Fig. 16. Definition diagram of 2D potential flow model.

i, j

i-1, j-1 i, j-1 i+1, j-1

i-1, j i+1, j

i-1, j+1 i, j+1 i+1, j+1

Fig. 17. Discretization of fluid domain.

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–22002196

To investigate the relationship between wave loads and waveperiod, we set the water depth at 6 m and performed analysesusing 1570 combinations of wave heights and wave periods.

Maximum value of breaking wave height was 4.68 m for waveswith period 7 s and longer. Therefore, we determined the range ofwave height as 0.01–4.6 m, with 0.1 m increment. Computedmaximum wave loads are plotted in Fig. 22 for selected waveheights. As shown in the plots, maximum vertical wave loadincreased with wave period; maximum horizontal wave loadpeaked at 4.4 s, then decreased with wave period. Wave lengthof 4.4 s wave was 27 m, about three times of the width of thebridge superstructure (9.4 m).

6.2. Wave loads vs. water depth

To investigate the effect of water depth on wave loads, weperformed additional analyses using different water depths. We

0

1000

2000

3000

4000

5000

0Wave Height (m)

Vert

ical

Wav

e Lo

ad (k

N)

1 2 3 4 5

2D HorizontalFlow-3D Horizontal 2D VerticalFlow-3D Vertical

Fig. 18. Comparison of maximum wave loads computed by 2D Potential Flow

model and Flow-3D.

Fig. 19. Comparison of maximum wave loads computed by 2D Potential Flow

model and large scale test results (Schumacher et al., 2008).

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–2200 2197

used 6 m as the reference water depth, and set water depth from5.6 to 6.6 m. The top surface of bridge superstructure was kept onSWL in these analyses. Maximum vertical wave loads weredivided by the maximum vertical load of 6 m water depth. Asshown in Fig. 23, maximum vertical wave loads decrease as waterdepth increases. However, the variation was not significant. Thevariation was also smaller at higher wave periods. Maximumhorizontal wave loads were less sensitive to water depth thancorresponding vertical wave loads.

6.3. Wave loads vs. bridge geometry

To investigate the effect of bridge superstructure geometry onwave loads, we varied the width of bridge superstructure from 6.9to 11.9 m. Results indicated that maximum vertical wave loadwas proportional to the width of bridge superstructures, whilemaximum horizontal load was not affected by the width of bridgesuperstructures (Meng, 2008).

7. Equations for calculation of maximum wave loads

We performed regression analysis using the results of parametricstudy to develop equations for calculation of maximum wave loads.Input parameters include wave height H, wave period T, water depth(including storm surge) d, bridge superstructure width Lx (alongwave direction), length Ly (across wave direction), depth Lz andclearance c (measured from top of superstructure to SWL, positivewhen top surface of the bridge superstructure is above SWL).

7.1. Reference case

The following equations were developed for the case in whichtop surface of the bridge superstructure was at SWL:

Fh0 ¼ rgHLyLzChðTÞ ð15Þ

ChðTÞ ¼ �0:13þ9:62

T�

30:80

T2þ

27:91

T3ð16Þ

Fv0 ¼ FbþrgHLxLyCvðTÞ ð17Þ

Cv Tð Þ ¼ 0:035þ0:21T�0:29ffiffiffiTp

for To3:6s ð18aÞ

Cv Tð Þ ¼ 0:79�0:053ffiffiffiTp�

1:73

Tfor TZ3:6s ð18bÞ

where Fh0 and Fv0 are, respectively, maximum horizontal andvertical wave loads, Fb is the buoyancy, r is the density ofseawater (1027 kg/m3 at 20 1C) and g is the acceleration due togravity (9.81 m/s2).

To include nonlinear wave effects, both horizontal and verticalwave load were multiplied by the ratio of wave crest height towave amplitude Zmax/A, where Zmax is the wave crest height and A

is the wave amplitude (H/2):

Fh0 ¼ 2rgZmaxLyLzCh Tð Þ ð19Þ

Fv0 ¼ Fbþ2rgZmaxLxLyCvðTÞ ð20Þ

7.2. Modification to structure elevation

For cases when the top surface of the bridge superstructurewas not at SWL, maximum horizontal wave load was modified asfollows:

FhðcÞ ¼ Fh0 �min 1,1�c�Zmax

Lz

� �ð21Þ

Maximum vertical wave load was modified as follows:

FvðcÞ ¼ Fv0 1�c

ZmaxþLz

� �for cZZmax�H ð22aÞ

FvðcÞ ¼ Fv0LzþH

LzþZmax

� �for coZmax�H ð22bÞ

Derivation of Eqs. (21) and (22) was explained in Section 5.When top surface of superstructure is above wave troughðcZZmax�HÞ, the vertical force Fv(c) is smaller than Fv0.

Fig. 20. Dimensions of 2D model for comparison with experimental results of Cuomo et al. (2007).

Fig. 21. Comparison of computed uplift forces on the deck element with

measurements of Cuomo et al. (2007).

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–22002198

7.3. Comparison with existing methods

We applied the equations developed in this section to calcu-late wave loads on the Biloxi Bay Bridge during Hurricane Katrina.Wave conditions during the Hurricane was hindcasted by Chenet al. (2009). Significant wave height was 2.6 m, correspondingmaximum wave height was 4.68 m and peak period was 5.5 s.Crest height according to Stokes 5th order wave theory was3.14 m. Clearance c was 1.28 m. Dimension of the superstructurewas given by Robertson et al. (2007) (see Table 1). Volume of thesuperstructure was 62.65 m3. Fh0 and Fv0 were calculated usingEqs. (19) and (20), respectively, Fh0¼843 kN and Fv0¼4220 kN.These results were then modified according to clearance c using

Eqs. (21) and (22). Because wave crest was higher than the topsurface, Fh(c) was equal to Fh0, while Fv(c)¼2943 kN. Weight ofthe superstructure was 1480 kN, which was not adequate to resistthe vertical wave load.

McConnell et al. (2004) proposed the following equation forestimation of quasi-static wave forces on structural componentsof a jetty structure:

Fqs

Fn¼

a

½Zmax�cl=Hs�b

ð23Þ

where Fqs is the quasi-static force, F* is the basic wave force, cl isthe distance from bottom of structure to SWL, Hs is the significantwave height and a and b are empirical coefficients determinedwith experimental data. Horizontal and vertical basis wave forcesFn

h and Fnv are determined by the product of pressure and projected

area of structural component.Horizontal and vertical wave loads on girders and deck

segments were calculated using the method proposed byMcConnell et al. (2004) (Tables 2 and 3). Total loads on the entiresuperstructure were 1142 and 3224 kN. The horizontal andvertical loads were, respectively, 36% and 10% higher than theresults of our method.

We also used equations in AASHTO (2008) to calculate themaximum vertical wave load. The vertical load was considered asthe sum of so-called ‘‘quasi-static’’ force and ‘‘slamming’’ force inAASHTO (2008). The ‘‘quasi-static’’ force and the ‘‘slamming’’ forcewere 3306 and 1616 kN, respectively, which gave a total verticalwave load of 4921 kN. This value was much higher than the result ofour method and the method of McConnell et al. (2004). Furtherexamination of the equations in AASHTO (2008) revealed that bridgewidth (Lx) was not explicitly included in the equation for vertical‘‘quasi-static’’ force. Results of our parametric study indicated thatmaximum vertical wave load was proportional to the width of

0

200

400

600

800

1000

1200

0Wave Period (s)

Max

. Hor

izon

tal W

ave

Load

(kN

)

H=0.5mH=1.0mH=1.5mH=2.0mh=2.5mh=3.0mh=3.5mh=4.0mh=4.5m

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Wave Period (s)

Max

. Ver

tical

Wav

eLoa

d (k

N)

H=0.5mH=1.0mH=1.5mH=2.0mh=2.5mh=3.0mh=3.5mh=4.0mh=4.5m

2 4 6 8 10 12

0 2 4 6 8 10 12

Fig. 22. Maximum wave loads vs. wave period. (a) Maximum Horizontal Wave

Load and (b) Maximum Vertical Wave Load.

0.98

0.99

1

1.01

1.02

5.4Water Depth (m)

Varia

tion

of M

ax. V

ertic

al W

ave

Load T=4 s

T=4.4 sT=4.8 sT=5.2 sT=5.6 sT=6 sT=6.4 sT=6.8 sT=7.2 sT=7.6 sT=8 s

5.6 5.8 6 6.2 6.4 6.6 6.8

Fig. 23. Variation of maximum vertical wave load vs. water depth.

Table 2Horizontal wave load on superstructures of Biloxi Bay Bridge using method of

McConnell et al. (2004).

Structural

component

Ly

(m)

Lz

(m)

cl

(m)

Fnh

(kN)

a b aZmax�cl

Hs

� �b Fh

(kN)

External beam 15.85 1.32 0.44 430 0.45 1.56 0.42 182

Internal beam 15.85 0.89 0.19 356 0.72 2.3 0.54 192

Total ¼183þ5�192¼1142 kN

Table 3Vertical wave load on superstructures of Biloxi Bay Bridge using method of

McConnell et al. (2004).

Structural

component

Lx

(m)

Ly

(m)

cl

(m)

Fnv

(kN)a b a

Zmax�clHs

� �b

Fv

(kN)

External beam 0.51 15.85 0.44 220 0.82 0.61 0.80 176

Internal beam 0.51 15.85 0.19 240 0.84 0.66 0.77 185

External deck 1.37 15.85 1.25 413 0.82 0.61 1.00 411

Internal deck 1.37 15.85 1.25 413 0.84 0.66 1.04 428Total ¼176þ5�185þ411þ4�428¼3224 kN

Table 1Wave loads on superstructures of Biloxi Bay Bridge using proposed method.

Hs

(m)

Lx

(m)

Ly

(m)

Lz

(m)

T

(s)

Ch

(T)

Cv

(T)

Fh0

(kN)

Fv0

(kN)

Fh

(kN)

Fv

(kN)

2.6 10.16 15.85 1.09 5.5 0.77 0.35 843 4220 843 2943

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–2200 2199

bridge superstructures; therefore, we do not recommend theequations in AASHTO (2008).

8. Summary and conclusions

In this paper, we used two numerical models to investigate thecomplicated problem of wave interaction with bridge superstruc-tures, and developed equations for calculation of wave loads onbridge superstructures. We first used Stokes wave theory to analyzekinematics of hurricane waves and concluded that method based onwave kinematics (Kaplan, 1992; Kaplan et al., 1995) should not beused to calculate wave loads on bridge superstructures. CFD soft-ware Flow-3D (Flow Science, 2008) was capable of modeling non-linear waves and wave breaking. We used Flow-3D to investigatethe effects of superstructure elevation and green water loading onwave loads. Due to the large computational time required by Flow-3D, we developed a 2D potential flow model. The model solvescomplex velocity potential from Laplace’s equation and boundaryconditions using finite difference method. Results of the 2D poten-tial flow model agreed well with experimental results and Flow-3Danalyses. Parametric study was then performed with the 2Dpotential flow model. A range of wave height, wave period, waterdepth and bridge geometry was covered in the parametric study.Equations for calculating wave loads on bridge superstructure werethen developed by regression analysis. Relationship between waveload and superstructure elevation established by Flow-3D analyseswas used to modify wave loads from the reference value to differentsuperstructure elevation. The equations we developed were appliedto calculate wave loads on Biloxi Bay Bridge during HurricaneKatrina. Comparison with results of the method of McConnellet al. (2004) and AASHTO (2008) indicated that both methods gavewave loads much higher than our method. The equations wedeveloped in this paper are easy to use. The equations can be usedin design of new coastal bridges and also in evaluation of vulner-ability of existing bridges to hurricane waves.

Acknowledgment

This study was supported in part by Texas Department ofTransportation (Grant no. 0-6063) and the Research AdvisoryCouncil of Texas A&M University at Galveston.

J. Jin, B. Meng / Ocean Engineering 38 (2011) 2185–22002200

References

AASHTO, 2008. Guide Specifications for Bridges Vulnerable to Coastal Storms.American Association of State Highway Transportation Officials, Washington, D.C.

ASCE TCLEE, 2006. Performance of Transportation Systems during HurricaneKatrina. Draft Report. American Society of Civil Engineers, Reston, Virginia.

API, 2000. Recommended Practice for Planning, Designing and Constructing FixedOffshore Platforms-Working Stress Design, API RP 2A-WSD. American Petro-leum Institute, Washington, D.C.

Bea, R.G., Xu, T., Stear, J., Ramos, R., 1999. Wave forces on decks of offshoreplatforms. J. Waterw., Port, Coastal, Ocean Eng. 125 (3), 136–144.

Bradner, C., 2008. Large-scale laboratory observations of wave forces on a highwaybridge superstructure. MS Thesis. Oregon State University, Corvallis, OR.

Buckley, B., 2006. Replacement of Florida’s escambia bay bridges on a fasttrack. Construct. Mag. /http://constructor.construction.com/features/build/archives/2006-09escambiaBridges.aspS.

Chen, Q., Wang, L., Zhao, H., 2009. Hydrodynamic investigation of coastal bridgecollapse during Hurricane Katrina. J. Hydraulic Eng. 135 (3), 175–186.

Cuomo, G., Tirindelli, M., Allsop, W., 2007. Wave-in-deck loads on exposed jetties.Coastal Eng. 54, 657–679.

Cuomo, G., Shimosako, K., Takahashi, S., 2009. Wave-in-deck loads on coastalbridges and the role of air. Coastal Eng. 56, 793–809.

Douglass, S.L., Chen, Q., Olsen, J.M., Edge, B.L., Brown, D., 2006. Wave forces onbridge decks. Final Report for U.S. Department of Transportation, FederalHighway Administration. Office of Bridge Technology, Washington, D.C.

Eamon, C.D., Fitzpatrick, P., Truax, D.D., 2007. Observations of structural damagecaused by Hurricane Katrina on the Mississippi Gulf Coast. J. Perform.Construct. Facilities 21 (2), 117–127.

Fenton, J.D., 1985. A fifth-order Stokes theory for steady waves. J. Waterw., Port,Coastal, Ocean Eng. 111 (2), 216–234.

Flow Science, Inc., 2008. Flow-3D User’s Manual, /www.flow3d.comS.Kaplan, P., 1992. Wave impact forces on offshore structures: re-examination and new

interpretations. In: Proceedings of the Offshore Technology Conference, Houston.Kaplan, P., Murray, J.J., Yu, W.C., 1995. Theoretical analysis of wave impact forces

on platform deck structures. In: Offshore Technology, vol. 1-A. OMAE Copen-hagen, Offshore Mechanics and Arctic Engineering Conference.

Lara, J.L., Garcia, N., Losada, I.J., 2006. RANS modeling applied to random waveinteraction with submerged permeable structures. Coastal Eng. 53 (5–6),395–417.

Le Mehaute, B., 1976. Introduction to Hydrodynamics and Water Waves. Springer-Verlag, New York.

Lin, P., Liu, P.L.F., 1998. A numerical study of breaking waves in the surf zone.J. Fluid Mech. 359, 239–264.

Marin, J., Sheppard, D.M., 2009. Storm surge and wave loading on bridge super-structures. In: Proceedings of the Structures 2009, ASCE, Austin, Texas.

McConnell, K., Allsop, W., Cruickshank, I., 2004. Wave forces on horizontalelements Piers, Jetties and Related Structures Exposed to Waves: Guidelinesfor Hydraulic Loading. Thomas Telford, London, pp. 60–63.

McPherson, R.L., 2008. Hurricane induced wave and surge forces on bridge decks.MS Thesis. Texas A&M University, College Station, TX.

Meng, B., 2008. Calculation of extreme wave loads on coastal highway bridges.Ph.D. Dissertation. Texas A&M University, College Station, TX.

Morison, J.R., O’Brien, M.P., Johnson, J.W., Schaaf, S.A., 1950. The force excerted bysurface waves on piles. Petrol. Trans., AIME 189, 149–154.

Ocean Engineering Associates (OEA), 2005. Hurricane Impact Analysis and Devel-opment of Design Criteria for the I-10 Bridges over Escambia Bay EscambiaCounty, Florida. Final Report. Florida Department of Transportation, Gaines-

ville, Fla.Okeil, A.M., Cai, C.S., 2008. Survey of short- and medium-span bridge damage

induced by Hurricane Katrina. J. Bridge Eng. 13 (4), 377–387.Padgett, J., DesRoches, R., Nielson, B., Yashinsky, M., Kwon, O., Burdette, N., Tavera,

E., 2008. Bridge damage and repair costs from Hurricane Katrina. J. Bridge Eng.13 (1), 6–14.

Panchang, V.G., Ge, W., Cushman, R.B., Pearce, B.R., 1991. Solution to the mild slopewave problem by iteration. Appl. Ocean Res. 13 (4), 187–199.

Rajagopalan, N., 2006. Bridge Superstructure. Alpha Science International Ltd.,Oxford, U.K.

Richardson, J.E., Panchang, V.G., 1998. Three-dimensional simulation of scour-inducing flow at bridge piers. J. Hydraul. Eng. 124 (5), 530–540.

Robertson, I.N., Riggs, H.R., Yim, S.C.-S., Young, Y.L., 2007. Lessons from HurricaneKatrina storm surge on bridges and buildings. J. Waterw., Port, Coastal, OceanEng. 133 (5), 463–483.

Schumacher, T., Higgins, C., Bradner, C., Cox, D., Yim, S., 2008. Large-scale waveflume experiments on highway bridge superstructures exposed to hurricane

wave forces. In: Proceedings of the 6th National Seismic Conference on Bridgesand Highways, Charleston, South Carolina.

Smith, H., Foster., D.L., 2005. Modeling of flow around a cylinder over a scouredbed. J. Waterw., Port, Coastal, Ocean Eng. 131 (1), 14–24.

Talbot, J., 2005. Repairing Florida’s Escambia Bay Bridge. Associated ConstructionPublications available online at /http://www.acppubs.com/article/CA511040.htmlS, March 2005.

USACE, 2002. Coastal Engineering Manual, Engineering Manual 1110-2-1100. U.S.Army Corps of Engineers, Washington, D.C (in 6 volumes).

Wamit Inc., 2008. WAMIT User Manual, Version 6.4. WAMIT Inc., Chestnut Hill.Weggel, J.R., 1972. Maximum breaker height. J. Waterw., Port, Coastal, Ocean Eng.

90 (WW4), 529–548.Zheng, Y.H., Shen, Y.M., Tang, J., 2007. Radiation and diffraction of linear water

waves by an infinitely long submerged rectangular structure parallel to avertical wall. Ocean Eng. 34 (1), 69–82.